Traffic Modelling and Related Queueing Problems

Presenter: Moshe Zukerman

ARC Centre for Ultra Broadband Information Networks

EEE Dept., The University of Melbourne

Presented at EE Dept., City University of Hong Kong, 11 April, 2002

Credit: R. Addie (USQ), T. Neame (EEE, Melbourne)

1. The big picture 2. Stream traffic modelling – ground rules3. Growth and efficiency4. Long Range Dependence (LRD) and LRD vs.

SRD + traffic models review 5. Poisson Pareto Burst Process (PPBP) 6. PPBP queueing performance7. PPBP fitting 8. Fundamentals and ideas9. The Resurrection of the MMPP

OUTLINE

The Big Picture

Traffic Modelling

Queueing Theory

PerformanceEvaluation

Simulations andFast Simulations

NumericalSolutions

Formulae inClosed Form

Traffic Measurements

Link and Network Design and Dimensioning

Traffic Prediction

Of Course, there are short cuts:

Just Do It! and then see …Pros and ConsBut let’s forget about these short cuts for now …

Research in Performance Evaluation

1. Exact analytical results (models)2. Exact numerical results (models)3. Approximations4. Simulations (slow and fast)5. Experiments6. Testbeds7. Deployment and measurements8. Typically, 4-7 validate 1-3.

Black Box

Parameters Performance

The black box, can have:Traffic (model or trace), and a system or a network model, and(1) Performance Formulae, or(2) Numerical solution, or(3) Fast Simulation, or(4) Simulation

Performance evaluation tool

Black Box

Parameters Performance

• Very Fast (micro-millisecond) for congestion control

• Fast (100s milliseconds) for Connection Admission Control

• Slow (days) for network design and dimensioning

How fast should the black box be?

Traffic Modelling

Ground Rules: For a given TT, S finds an SP defined by a small number of parameters such that:

(1)TT and SP give the same performance when fed into an SSQ for any buffer size and service rate.

(2)TT and SP have the same mean and autocorr.(3)Preferably, SP SSQ is amenable to analysis.

TrafficTrace (TT)

StochasticProcess (SP)

S

“Do Not” Rules

• We Do not take retransmissions into account.

• We ignore TCP dynamics.• The aim is to dimension network so that

retransmissions will normally not be needed.

• This will be efficient as traffic, capacity and number of users increase. (why??)

Why does efficiency increase with growth?

• If load and capacity increases service rate is better- Scaling (Frank Kelly).

• Convergence to Gaussian - Central Limit Theorem (Addie) – but the convergence is slow.

),(),(max)( tttStttAtQt

Reich’s Formula:

Consider two scenarios (1 and 2). Let:A1(s,t)= A2(bs,bt) S1(t-t,t)=b S2(t-t,t)= S2(b(t-t),bt)

Q(t) = queue size at time tA(s,t)=Work arrive between s and tS(s,t)=Service capacity between s and t

Scaling

)(

)),((),(max

)),(()),((max

),(),(max)(

2

22

22

111

btQ

bttbbtSbttbbtA

btttbSbtttbA

tttStttAtQ

t

t

t

Thus, queue size is the same but served much faster!

LRD Convergence to Gaussian

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

0 500 1000 1500 2000

Buffer size, x

Pr(

Q >

x)

Gaussian

Why does LRD traffic converge to Gaussian slowly?

I will tell you later.First, let me explain the meaning of

LRD traffic and the difference between LRD and SRD.

Background• Modelling packet traffic is difficult because of its

properties• Long Range Dependence (LRD) is a widespread

phenomenon• LRD has been found in

– Ethernet traffic (Leland, et al. 94)– VBR video traffic (Garrett & Whitt 94)– Internet traffic (Paxson 95)– MAN traffic (Zukerman, et al. 95)– ATM cell traffic (Jerkins & Wang 97)– CCS Signalling Traffic (Duffy, et al. 94)

• Possible causes for LRD– Distribution of file sizes– Human behaviour– VBR video

Long Range Dependence

• A process is defined to be Long Range Dependent if its autocorrelation function R() decays slower than exponentially.

• LRD need only exist in the limit – LRD implies nothing about its short term correlations

which affect performance in small buffers

• Traditional traffic model processes are Short Range Dependent (SRD)

• In practice, LRD is difficult to distinguish from non-stationarity

•1986 Heffes and Lucantoni IEEE JSAC IEEE Best Paper Award (MMPP)•1994 Leland et al. ACM/IEEE TON IEEE Best Paper Award (LRD)•1995 Likhanov et al. INFOCOM Best Paper Award – proposed a model equivalent to the one we call Poisson Pareto Burst Process (PPBP)

Traffic Modeling: Historical Highlights

-5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0

The Variance

v=Autocovariance Sum = The variance

Arrival Process Autocovariance (IID)

-5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0

The Variance

v=Autocovariance Sum

Arrival Process Autocovariance SRD

For SRD

v = limn VAR [A(n)]/n

That is, for large n,VAR [A(n)] grows linearly with n.

-5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0

The Variance

V = Autocovariance Sum =

Arrival Process Autocovariance (LRD)

For LRD

1,>with

)]([VAR

CnnA

LRD

SRD

IID

BUFFER SIZE

LO

SS

PR

OB

AB

ILIT

Y

Queueing Performance Comparison

Self Similar Traffic (Ethernet) 1 Second Intervals 10 Second Intervals 60 Second Intervals

100 Second Intervals Half Hour Intervals 1 Hour Intervals

Arrivals PoissonInterval = 1

30

40

50

60

70

80

0 50 100 150

Interval = 10

300

400

500

600

700

0 50 100 150

Interval = 100

3000

4000

5000

6000

7000

0 50 100 150

Interval = 1000

30000

40000

50000

60000

70000

0 50 100 150

For Poisson (IID) or SRD

VAR A t t

A tA t

tt

[ ( )]

[ ( )][ ( )]

For Poisson = 1,

Burstiness = SDE

For LRD

VAR

with > 1,

Burstiness = SDE

[ ( )]

[ ( )][ ( )]

A t t

A tA t

tt

LRD vs. SRD

Slope=1: Non-fractal (SRD)

Slope>1: Fractal (LRD)

Log V(A(t))

Log (t)

The Hurst Parameter

lim [ ( )]t A t t

H

H H

VAR

with, 2 > 0 (LRD: 2 > > 1)

The Hurst Parameter ( ):

= / 2, (LRD: 1 > > 1/ 2)

Burstiness on ALL scales? Not Really.

1)2/()]([ E

)]([ SD

1>>2with

)]([VAR

CttA

tA

ttA

Still approaches zero as t grows because < 2

MMPP (SRD)

SRD Gaussian LRD Gaussian

PPBP (LRD)

QueueingAnalysis

Done SSQ

Approx.

(AZ 92,93,94)

SSQ

Approx. (AZN 95)

Useless bounds + results here

Fast/Slow

Simulat’n

Fast + slow

Fast + slow Fast Only Slow + results here

Queueing PerformanceState of the art

The Markov Modulated Poisson Process (MMPP) ?

• Several traffic activity modes• The period of each activity mode has exponential distribution

• When in mode i – Poisson iTime

Two State MMPP Results

• Analytical results for queueing performance are available•Four parameters•Fitted to mean, variance, v, + •Numerical algorithms (Matrix Geometric) results available for n state MMPP.

Gaussian Queueing Results

• SRD: results as a function of three parameters which can be fitted to mean, variance, v•LRD: results as a function of three parameters which can be fitted to mean, variance, H.

Exponential Pareto

Poisson Pareto Burst Process (PPBP)• Total work from multiple overlapping bursts

• Burst arrivals according to a Poisson process of rate

• Burst durations are Pareto distributed

• During a burst, bit rate is constant

• All bursts have constant bit rate

Pareto Distribution

• For the Pareto distribution– Is a heavy-tailed distribution– Has infinite variance, but finite mean

,Pr{ }

1, otherwise

xx

X x

1)(

XE

21

Why the PPBP?

• Assume each user transmits at maximum capacity, or zero capacity– Each user can be represented as an on-off process

• Assume users are independent of one another• Assume duration of on times is heavy-tailed• The PPBP is a limiting case for the sum of multiple

independent heavy-tailed on-off processes – shown in Likhanov, et al. 95

Heavy Tailed Distribution

• Complementary distribution functions for exponential distribution and Pareto distribution with infinite variance. Both distributions have mean = 3

0.001

0.01

0.1

1

0 2 4 6 8 10 12 14 16 18 20

xP

r{x

> x

}

Exponential

Pareto

The Poisson Pareto Burst Process (PPBP)

An is total work arriving in a fixed size interval of length t

Exponential ()Pareto (, )

r

n

An

PPBP Parameter Fitting

1)(

r

AE n

21

2

3 H

6/1))1(

),(

2/((

2

22

2

)(

Fr

rnAVar

11

)3)(2)(1()2(2)3(6),(

23

F

Recall: LRD Convergence to GaussianTo fit we choose the best curve!

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

0 500 1000 1500 2000

Buffer size, x

Pr(

Q >

x)

Gaussian

Why does LRD traffic converge to Gaussian slowly?

I can tell you now.

time period

Long Bursts Short Bursts:A very good way to consider LRD

traffic

PPBP Initial Conditions

• “Steady state” process has Poisson number of active bursts at time t

• Mean number of bursts is E(D)

• Remaining duration in each burst distributed according to its forward recurrence time

Pareto Forward Recurrence Times

1 2

• Even heavier tail than ordinary Pareto

• Has infinite mean and infinite variance

xx

xx

xR1

11

,1

}Pr{

1

Forward Recurrence Times – Even Heavier Tails

• Complementary distribution functions for Pareto distribution with mean 3, and corresponding forward recurrence times

Long Bursts and Short Bursts

• Consider a PPBP over the interval (t, t + W)• At time t, there will be Bt bursts active• Each of these Bt bursts will last the entire interval with

probability

• Label the bursts present at the start and the end of the interval as long bursts

• Rest of the bursts are the short bursts

11

}Pr{W

WR

t t+W

Long burst process Short burst process

t+Wt t t+W

Long Burst – Short Burst Division

Properties• Long bursts and short bursts processes are independent

of one another.• Number of long bursts, n, is Poisson distributed with

mean E(D)Pr(R > W)• For a given interval, long bursts process is a CBR

component• Short bursts process is a correlated Poisson process

with mean rE(D)(1-Pr(R > W))• Short bursts process is not LRD

Single Server Queue• In interval n:

– An is the number of cells arriving

– C is the fixed service rate

– Yn is the net arrival process

– Qn is the amount of work in the buffer

CAY nn

1n n nQ Q Y

0,max XX

A nnVQn

CAn

Our Queueing Performance Results• Overflow probability for short burst process

depends on n, number of long bursts

• Define S(x,n) as overflow probability when number of long bursts is n

• Probability of n long bursts is Poisson with mean E(D)Pr(R > W)

• Overall overflow probability for PPBP is:

0

),(}Pr{}Pr{n

nxSnNxQ

Instability in a Simulation• Probability of n long bursts is Poisson with mean E(d)Pr(R > W)

• n long bursts leaves capacity C – nr for short bursts

• Mean arrival rate for short bursts process is r E(d)(1-Pr(R > W))• There is non-zero probability that the system will

be unstable for the duration of the simulation• Can choose simulation length, W, such that this

probability is negligible• Have demonstrated that overall impact of

instability is limited

Simulation with Random Number of Long Bursts

• Long bursts may have significant impact on simulation results

• Initialise simulation with a random number of bursts and let a random number of these be long bursts

• Will require a large number of simulations to be sure that the state space is explored thoroughly

Weighted Sum to Account for Long Bursts

• Create a process with no long initial bursts

• Simulate and find losses in systems with rate C-nr

• Sum the losses from these systems, weighted by the probability that N = n

Effect of Improved Simulations

0.0001

0.001

0.01

0.1

1

0 50000 100000 150000 200000

Buffer threshold, x

Pr{Q >

x}Weighted Sum Simulation

Simple Simulation

IP trace fittingW = 22,000,000 x 60 simulations

Simple simulation

Quasi-Stationary Estimate

• Long burst process is constant for period W• Use existing techniques to estimate overflow

probabilities for short burst process fed into server with capacity C - nr

• As with simulation, combine estimates according to probabilities of n long bursts

• Which W gives best estimate?– Choose the W which gives highest overflow probability.

Quasi-Stationary (QS) Estimates

W

W W

W

QS

Est

imat

e fo

r W

QS

Est

imat

e fo

r W

QS

Est

imat

e fo

r W

QS

Est

imat

e fo

r W

Large Deviations Results

• Large deviations is a good approach for Gaussian queues.

• It has failed to provide useful results for LRD non Gaussian queues.

• There was an attempt to use Large deviation to obtain analytical results for the continuous time counterpart of the queue fed by PPBP (called the M/G/ process) by Tsybakov & Georganas.

• We will discuss it now.

Large Deviations Results (cont.)

• Large deviations results for queues fed by LRD sources have been derived by a number of authors.– Results only hold as , i.e. for large

buffers

• Most useful results for M/G/ input are due to Tsybakov & Georganas– They give upper and lower bounds:

x

kk BxxQAx )1()1( }Pr{

)(1 dE

rC

k

Large Deviations Results (cont.)

• A and B are constant with respect to the buffer size x.

• Note that the upper and lower bounds have the same form.

• We will compare our results against these bounds.

kk BxxQAx )1()1( }Pr{

)(1 dE

rC

k

Comparison of ResultsProb

{Q >

}

Second Set of ResultsProb

{Q >

}

Quasi-Stationary Estimate

• Previously have used repeated simulation to fit final parameter

• Using the estimate, fitting is faster and more reliable

• Quasi-stationary value is still only an estimate– Most accurate when is large– for PPBP fitted to real data is usually small

Reliability is questionable

Modelling Measured Traffic

• PPBP has 4 parameters– Burst arrival rate – Rate of work per burst r– 2 Pareto parameters, and

• Parameter Fitting– Can fit r, , to measured mean, variance, H– Can produce PPBPs with same r, , values which

give very different queueing performance resultsFitting is vital to give a model which predicts the

performance of real traffic

PPBP Convergence to Gaussian

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

0 500 1000 1500 2000

Buffer size, x

Pr(

Q >

x)

Gaussian

Good News! - Fitting the PPBP to Real Traffic

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

0 20000 40000 60000 80000 100000

Buffer threshold, x (packets)

Pr(

Q >

x)

IP Trace Gaussian

0.00E+00

5.00E+06

1.00E+07

1.50E+07

2.00E+07

2.50E+07

0 500 1000 1500 2000 2500 3000

Lag, k

Aut

ocov

aria

nce

PPBP model (from analysis)

PPBP model (from simulation)

Trace

Autocovariance Fitting

How Are We Doing?

• Our aim was to find a stochastic process with the following properties:Defined by a small number of parameters

• PPBP has only 4 parameters

– If the parameters of the process are fitted using measurable statistics of an actual traffic stream then

• 3 of 4 parameters fitted to measurable statistics• Fitting the 4th parameter can now be done systematicallyThe first and second order statistics of the model will match

those of the traffic stream

How Are We Doing? (cont.)

If fed through an SSQ the performance results of the model will accurately predict those of the real traffic stream in an identical SSQ. This should be true for a wide range of buffer sizes and service rates.

•PPBP does wellIf performance results can be calculated analytically, so much the better

•Quasi-stationary estimate provides an accurate analytic estimate

Summary (PPBP)

• Described the Poisson Pareto Burst Process (PPBP)• Long bursts may have a significant impact on PPBP

simulation results.• We can factor long bursts into simulations.• Quasi-stationary analysis gives a reasonable estimate

of the queueing performance of the PPBP.• Using quasi-stationary estimate, reliable fitting of the

PPBP to real traffic streams is possible.

Speculation: MMPP ResurrectionAnti-thesis??

• The PPBP at this stage is not amenable to state dependent queueing analysis.

• Such analysis is needed in many Telecommunications systems.

• MMPP or other Markovian models are amenable to such analyses.

• I will present now intuitive arguments for the resurrection of MMPP.

Single Server Queue Realization

G/G/1

Unfinished Work Distribution

Reich’s Formula (1958)

P V T P A B Tw

ii

w

( ) max ( )

1

1

Ai= work arrives during interval iB= Service capacity during interval i

Again, our SSQ Realization:The relevant correlation duration is from the beginning of the last busy period.Indeed, LRD Longer busy periods.

G/G/1

BUFFER

But if we introduce a buffer …???

G/G/1/2

The buffer breaks long busy periods to many short ones!

And we may not need to consider LRD in our traffic modelling.

Queueing Performance Comparison

LRD

SRD

IID

BUFFER SIZE

LO

SS

PR

OB

AB

ILIT

YL

OG

LRD

SRD

IID

BUFFER SIZELO

SS

PR

OB

AB

ILIT

Y

The resurrection: SRD (MMPP?) as a model for LRDL

OG

LRD vs. SRD

Slope=1: Non-fractal (SRD)

Slope>1: Fractal (LRD)

Log V(A(t))

Log (t)

The resurrection:SRD (MMPP?) as a model for LRD

Slope=1: Non-fractal (SRD)

Slope>1: Fractal (LRD)

Log V(A(t))

Log (t)

There are arguments for resurrection of the MMPP asA traffic model.

Conclusion

Two approaches:(1)PPBP (State Dependent (?)) (2)MMPP (Accurate traffic model (?))

The Game goes on